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On the Hardy--Littlewood--Chowla conjecture
An old conjecture dating back at least to Hildebrand's work in 1989, asserts that the Möbius function exhibits cancellation on shifted primes; that is, $$\sum_{p\le x}μ(p+2)=o(π(x)),\quad \text{or equivalently}\quad \sum_{n\le x}Λ(n)μ(n+2)=o(x).$$ This is also a well-known model case for the parity
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